New insight into results of Ostrowski and Lang on sums of remainders using Farey sequences

The sums \(S(x,t)\) of the centered remainders \(kt-\lfloor kt\rfloor - 1/2\) over \(k \leq x\) and corresponding Dirichlet series were studied by A. Ostrowski, E. Hecke, H. Behnke and S. Lang for fixed real irrational numbers \(t\). Their work was originally inspired by Weyl's equidistribution...

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Bibliographic Details
Published in:arXiv.org 2021-03
Main Author: Kunik, Matthias
Format: Article
Language:English
Subjects:
Dirichlet problem
Irrationality
Number theory
Scaling
Sequences
Sums
Online Access:Get full text
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Summary:The sums \(S(x,t)\) of the centered remainders \(kt-\lfloor kt\rfloor - 1/2\) over \(k \leq x\) and corresponding Dirichlet series were studied by A. Ostrowski, E. Hecke, H. Behnke and S. Lang for fixed real irrational numbers \(t\). Their work was originally inspired by Weyl's equidistribution results modulo 1 for sequences in number theory. In a series of former papers we obtained limit functions which describe scaling properties of the Farey sequence of order \(n\) for \(n \to \infty\) in the vicinity of any fixed fraction \(a/b\) and which are independent of \(a/b\). We extend this theory on the sums \(S(x,t)\) and also obtain a scaling behaviour with a new limit function. This method leads to a refinement of results given by Ostrowski and Lang and establishes a new proof for the analytic continuation of related Dirichlet series. We will also present explicit relations to the theory of Farey sequences.
ISSN:2331-8422